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# help

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In a University out of 120 students, 15 opted mathematics only, 16 opted statistics only, 9 opted physics only and 45 opted physics and mathematics, 30 opted physics and statistics, 8 opted mathematics and statistics, and 80 opted physics.

Find the sum of number of students who opted mathematics and those who didn't opted any of the subjects given.

Jun 7, 2020

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In a University out of 120 students, 15 opted mathematics only, 16 opted statistics only, 9 opted physics only and
45 opted physics and mathematics, 30 opted physics and statistics, 8 opted mathematics and statistics, and
80 opted physics.
Find the sum of number of students who opted mathematics and those who didn't opted any of the subjects given. My attempt:

$$\begin{array}{|rcll|} \hline x+y+t+9 &=& 80 \quad | \quad x+y = 45 \\ 45+t+9 &=& 80 \\ t+54 &=& 80 \\ t &=& 80-54 \\ \mathbf{ t } &=& \mathbf{26} \\ \hline \end{array} \begin{array}{|rcll|} \hline 30 &=& y+t \\ -~~8 &=& y+z \\ \hline 22 &=& y+t-(y+z) \\ 22 &=& y+t-y-z \\ 22 &=& t-z \\ z+22 &=& t \\ \mathbf{ z } &=& \mathbf{t-22} \quad | \quad \mathbf{ t =26} \\ z &=& 26-22 \\ \mathbf{ z } &=& \mathbf{4} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline z+y &=& 8 \\ y &=& 8-z \quad | \quad \mathbf{z=4} \\ y &=& 8-4\\ \mathbf{ y } &=& \mathbf{4} \\ \hline \end{array} \begin{array}{|rcll|} \hline x+y &=& 45 \\ x &=& 45-y \quad | \quad \mathbf{y=4} \\ x &=& 45-4\\ \mathbf{ x } &=& \mathbf{41} \\ \hline \end{array}$$

The sum of number of students who opted mathematics:

$$\begin{array}{|rcll|} \hline \text{Mathematics} &=& 15+x+y+z \\ \text{Mathematics} &=& 15+41+4+4 \\ \mathbf{\text{Mathematics}} &=& \mathbf{64} \\ \hline \end{array}$$

The sum of number of students who didn't opted any of the subjects given:

$$\begin{array}{|rcll|} \hline \text{didn't opted any} &=& 120-(9+15+16+x+y+z+t) \\ \text{didn't opted any} &=& 120-(40+41+4+4+26) \\ \text{didn't opted any} &=& 120-115 \\ \mathbf{\text{didn't opted any}} &=& \mathbf{5} \\ \hline \end{array}$$ Jun 8, 2020